Arens-Michael enveloping algebras and analytic smash products

Abstract

Let g be a finite-dimensional complex Lie algebra, and let U(g) be its universal enveloping algebra. We prove that if \hat{U}(g), the Arens-Michael envelope of U(g), is stably flat over U(g) (i.e., if the canonical homomorphism U(g)-->\hat{U}(g) is a localization in the sense of Taylor), then g is solvable. To this end, given a cocommutative Hopf algebra H and an H-module algebra A, we explicitly describe the Arens-Michael envelope of the smash product A#H as an ``analytic smash product'' of their completions w.r.t. certain families of seminorms.